We prove a quantitative version of the Gibbard-Satterthwaite theorem. We show that a uniformly chosen voter profile for a neutral social choice function f of q ≥ 4 alternatives and n voters will be manipulable with probability at least 10−4∈2 n −3 q −30, where ∈ is the minimal statistical distance between f and the family of dictator functions. Our results extend those of [11], which were obtained for the case of 3 alternatives, and imply that the approach of masking manipulations behind computational hardness (as considered in [4,6,9,15,7]) cannot hide manipulations completely. Our proof is geometric. More specifically it extends the method of canonical paths to show that the measure of the profiles that lie on the interface of 3 or more outcomes is large. To the best of our knowledge our result is the first isoperimetric result to establish interface of more than two bodies.